Lifting Tropical Curves in Space and Linear Systems on Graphs

Details Lifting Tropical Curves in Space and Linear Systems on Graphs Lifting Tropical Curves in Space and Linear Systems on Graphs

2010-09-09

arxiv.org/abs/1009.1783v3

Mathematics

Algebraic Geometry

substantial revision

Scientific paper

Tropicalization is a procedure for associating a polyhedral complex in Euclidean space to a subvariety of an algebraic torus. We study the question of which graphs arise from tropicalizing algebraic curves. By using Baker's specialization of linear systems from curves to graphs, we are able to give a necessary condition for a balanced weighted graph to be the tropicalization of a curve. Our condition reproduces a generalization of Speyer's well-spacedness condition and also gives new conditions. In addition, it suggests a new combinatorial structure on tropicalizations of algebraic curves.

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